Method for determining anisotropic resistivity and dip angle in an earth formation

ABSTRACT

A new method for determining the anisotropic resistivity properties of a subterranean earth formation traversed by a borehole utilizing a multi-component induction logging tool. The method utilizes the phase and attenuation signals induced by eddy currents in the formation to determine the resistive properties, with or without prior knowledge of the borehole dip angle.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention is related to the field of electromagnetic logging of earth formations penetrated by well borehole. More specifically, the invention is related to a method for determining the anisotropic resistivity properties of the earth formation and the dip angle of the borehole in the earth formation.

[0003] 2. Description of the Related Art

[0004] The basic techniques of electromagnetic or induction logging instruments are well known in the art. A sonde, having at least one transmitter coil and at least one receiver coil, is positioned in a well borehole either on the end of a wireline or as part of a logging while drilling (“LWD”). The axis of the coils is essentially co-linear with the axis of the sonde and borehole. An oscillating signal is transmitted through transmitter coil, which creates a magnetic field in the formation. Eddy currents are induced in the earth formation by the magnetic field, modifying the field characteristics. The magnetic field flows in ground loops essentially perpendicular to the tool axis and is picked up by the receiver coil. The magnetic field induces a voltage in the receiver coil related to the magnitude of the earth formation eddy currents. The voltage signals are directly related to the conductivity of the earth formation, and thereby conversely the formation resistivity. Formation resistivity is of interest in that one may use it to infer the fluid content of the earth formation. Hydrocarbons in the formation, i.e. oil and gas, have a higher resistivity (and lower conductivity) than water or brine.

[0005] However, the formation is often not homogeneous in nature. In sedimentary strata, electric current flows more easily in a direction parallel to the strata or bedding planes as opposed to a perpendicular direction. One reason is that mineral crystals having an elongated shape, such as kaolin or mica, orient themselves parallel to the plane of sedimentation. As a result, an earth formation may posses differing resistivity/conductivity characteristics in the horizontal versus vertical direction. This is generally referred to as formation microscopic anisotropy and is a common occurrence in minerals such as shales. The sedimentary layers are often formed as a series of conductive and non-conductive layers. The induction tool response to this type of formation is a function of the conductive layers where the layers are parallel to the flow of the formation eddy currents. The resistivity of the non-conductive layers is represents a small portion of the received signal and the induction tool responds in a manner . However, as noted above, it is the areas of non-conductivity (high resistivity) that are typically of the greatest interest when exploring for hydrocarbons. Thus, conventional induction techniques may overlook areas of interest.

[0006] The resistivity of such a layered formation in a direction generally parallel to the bedding planes is referred to as the transverse or horizontal resistivity R_(h) and its inverse, horizontal conductivity σ_(h). The resistivity of the formation in a directive perpendicular to the bedding planes is referred to as the longitudinal or vertical resistivity R_(v), with its inverse vertical conductivity σ_(v). The anisotropy coefficient, by definition is: $\begin{matrix} {\lambda = {\sqrt{R_{H}/R_{V}} = {\sqrt{\sigma_{v}/\sigma_{h}} = \frac{1}{\alpha}}}} & \lbrack 1\rbrack \end{matrix}$

[0007] Subterranean formations are often made up of a series of relatively thin beds having differing lithological characteristics and resistivities. When the thin individual layers cannot be delineated or resolved by the logging tool, the logging tool responds to the formation as if it were macroscopically anisotropic formation, ignoring the thin layers.

[0008] Where the borehole is substantially perpendicular to the formation bedding planes, the induction tool responds primarily to the horizontal components of the formation resistivity. When the borehole intersects the bedding planes at an angle, often referred to as a deviated borehole, the tool will respond to components of both the vertical and horizontal resistivity. With the increase in directional and horizontal drilling, the angle of incidence to the bedding planes can approach 90°. In such instances, the vertical resistivity predominates the tool response. It will be appreciated that since most exploratory wells are drilled vertical to the bedding planes, it may be difficult to correlate induction logging data obtained in highly deviated boreholes with known logging data obtained in vertical holes. This could result in erroneous estimates of formation producibility if the anisotropic effect is not addressed.

[0009] A number of techniques and apparatus have been developed to measure formation anisotropy. These techniques have included providing the induction tool with additional transmitter and receiver coils, where the axes of the additional coils are perpendicular to the axes of the conventional transmitter and receiver coils. An example of this type of tool might include U.S. Pat. No. 3,808,520 to Runge, which proposed three mutually orthogonal receiver coils and a single transmitter coil. Other apparatus include the multiple orthogonal transmitter and receiver coils disclosed in U.S. Pat. No. 5,999,883 to Gupta et al. Still other techniques have utilized multiple axial dipole receiving antennae and a single multi-frequency transmitter, or multiple axial transmitters such as those described in U.S. Pat. No. 5,656,930 to Hagiwara and U.S. Pat. No. 6,218,841 to Wu.

SUMMARY OF THE INVENTION

[0010] A new method is provided for determining the anisotropic properties of a subterranean earth formation. The present invention is directed to a method for determining the anisotropic properties of an earth formation utilizing a multi-component induction. Specifically, the present invention contemplates a method for inverting the multi-component induction tool responses to determine anisotropic resistivity of an anisotropic and/or homogeneous formation and determine the tool's orientation with respect to the formation anisotropic direction utilizing both the resistive (R) and reactive (X) portions of the signals from a combination of tool responses.

[0011] In a preferred implementation, an induction logging tool, having multiple mutually orthogonal transmitter coils and receiver coils, is positioned in a borehole and activated. Power is applied to the transmitter coils to induce eddy currents in the formation. These eddy currents then induce currents in the receiver coils. These are processed to generate a preliminary phase shift derived resistivity and attenuation derived resistivity. This information is then compared with a predetermined model that relates phase shift derived resistivity and attenuation derived resistivity, horizontal resistivity, vertical resistivity and the anisotropy coefficient. Utilizing an inversion technique based on the preexisting model, the horizontal resistivity and vertical resistivity for a formation, as well as anisotropy coefficient and deviation angle relative to the formation, may be readily determined from the logging data.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012] A better understanding of the present invention may be had by referencing detailed description together with the Figures, in which:

[0013]FIG. 1 is simplified depiction of a logging tool that may be used to practice the present invention deployed in an earth borehole;

[0014]FIG. 2A depicts the orientation of the tool transmitters and receivers;

[0015]FIG. 2B depicts the relationship between the borehole, formation and tool coordinate systems;

[0016]FIG. 3A is a nomograph depicting a means for determining R_(H) as a function of R_(ll);

[0017]FIG. 3B is a nomograph depicting a means for determining R_(H) as a function of R_(ll) and X_(ll);

[0018]FIG. 3C is a nomograph depicting a means for determining R_(V)/R_(H) as a function of R_(tt);

[0019]FIG. 3D is a nomograph depicting a means for determining R_(V)/R_(H) as a function of the ratio R_(tt)/R_(ll);

[0020]FIG. 4A is a nomograph depicting a means for determining R_(H) and R_(V) as a function of R_(tt) and X_(tt);

[0021]FIG. 4B is a nomograph depicting a means for determining R_(V)/R_(H) as a function of the ratio R_(tt)/X_(tt);

[0022]FIG. 5A is a nomograph depicting a means for determining R_(H) as a function R_(tt);

[0023]FIG. 5B is a nomograph depicting a means for determining R_(H) as a function of R_(tt) and X_(tt);

[0024]FIG. 5C is a nomograph depicting a means for determining R_(V)/R_(H) as a function of R_(ll);

[0025]FIG. 5D is a nomograph depicting a means for determining R_(V)/R_(H) as a function of R_(uu);

[0026]FIG. 5E is a nomograph depicting a means for determining R_(V)/R_(H) as a function of the ratio of R_(ll)/R_(tt);

[0027]FIG. 5F is a nomograph depicting a means for determining R_(V)/R_(H) as a function of the ratio of R_(uu)/R_(tt);

[0028]FIG. 6A is a nomograph depicting a means for determining R_(H) and R_(V) as a function of R_(ll) and X_(ll);

[0029]FIG. 6B is a nomograph depicting a means for determining R_(V)/R_(H) as a function of the ratio of X_(ll)/R_(ll);

[0030]FIG. 6C is a nomograph depicting a means for determining R_(H) and R_(V) as a function of R_(uu) and X_(uu);

[0031]FIG. 6D is a nomograph depicting a means for determining R_(V)/R_(H) as a function of the ratio of X_(uu)/R_(uu);

[0032]FIG. 7A is a nomograph depicting a means for determining R_(H) and β as a function of R_(uu) and X_(ll); and

[0033]FIG. 7B is a nomograph depicting a means for determining R_(H) as a function of the ratio of R_(ll)/X_(ll).

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0034] The present invention is intended to be utilized with a multi-component, i.e., multiple mutually orthogonal transmitters and receiver coils. Exemplary of this type of induction tool is that disclosed in U.S. Pat. No. 5,999,883 to Gupta et al., which is incorporated herein by reference. In FIG. 1, an induction tool 10 is disposed in a wellbore 2 drilled through an earth formation 3. The earth formation 3 is shown as having a zone of interest 4. The tool 10 is lowered into the earth formation 3 to the zone of interest 4 on an armored, multi-conductor cable 6. The cable 6 is further part of a surface system (not shown) which might typically consist of a winch, a surface control system, including one or more surface computers, interface equipment, power supplies and recording equipment. The surface systems of this type might include a mobile truck mounted unit or a skid mounted unit for offshore operations. Further, the tool 10 may be transported utilizing other techniques, such as coiled tubing having power and data communications capability or as part of a drilling string in a Logging While Drilling (LWD) suite of tools.

[0035] The tool 10 is comprised of three subsections including an electronics section 14, a coil mandrel unit 8, and a receiver/processing telemetry section 12 that is in communication with the cable 6. The coil mandrel section 8 includes the transmitter coils for inducing an electromagnetic field in the earth formation in the zone of interest 4 upon application of power and the receiving coils for picking up signals created by induced eddy currents characteristic is the zone of interest 4. The electronics section 14 include the signal generator and power systems to apply the current to the transmitter coils. The tool 10 is shown as being disposed adjacent to a zone of interest 4 that is made up of thin formation sections 4A-4E.

[0036] It should be noted that while FIG. 1 depicts the tool 10 as being lowered in a vertical borehole 2, that current drilling techniques commonly result in a borehole which deviates several times along its length from the true vertical position. Accordingly, a borehole may intersect a zone of interest at an angle and could greatly affect the tool's measurement of resistivity characteristics. The preferred method of the present invention is designed to address this problem.

[0037] 1. Relation of Tool, Borehole and Formation Coordinate Systems

[0038] A multi-component induction tool such as that disclosed in U.S. Pat. No. 5,999,883 consists of at least three mutually orthogonal loop antenna transmitters (M_(l), M_(m), M_(n)) and at least three mutually orthogonal receiver coils whose responses are proportional to the magnetic field strength vectors (H_(l), H_(m), H_(n)), where l, m, and n denote a common coordinate system. It should be noted that the in-phase R-signal is proportional to the imaginary part of the H field. The three transmitters are set at the same position longitudinally along the tool 10 axis, which coincides with the l-axis of the coordinate system. The three receivers are also grouped at a common position spaced away from the transmitters along the l-axis. The two transverse directions are along the m- and n-axes.

[0039] With this arrangement, there exist nine different complex voltage measurements proportional to the magnetic field strength vector at the receiver loop antennas when the transmitters are activated:

[0040] (H_(ll), H_(ml), H_(nl)) from transmitter M_(l);

[0041] (H_(lm), H_(mm), H_(nm)) from transmitter M_(m); and

[0042] (H_(ln), H_(mn), H_(nn)) from transmitter M_(n).

[0043] However, as a matter of reciprocity, H_(nl)=H_(ln); H_(mn)=H_(nm); and H_(ml)=H_(lm). Accordingly, there are six independent measurements, which can be expressed as: $\begin{matrix} {H^{tool} = {\begin{bmatrix} H_{ll} & H_{l\quad m} & H_{l\quad n} \\ H_{m\quad l} & H_{m\quad m} & H_{mn} \\ H_{nl} & H_{n\quad m} & H_{nn} \end{bmatrix} = \begin{bmatrix} H_{ll} & H_{l\quad m} & H_{l\quad n} \\ H_{l\quad m} & H_{m\quad m} & H_{mn} \\ H_{l\quad n} & H_{mn} & H_{nn} \end{bmatrix}}} & \lbrack 2\rbrack \end{matrix}$

[0044] One starts with the assumption that formations 4A-4E of FIG. 1 are layered horizontally, where the true vertical direction is z-axis. A formation will be said to exhibit anisotropy where the resistivity in the vertical direction is different from that in the horizontal direction. The formation conductivity tensor is characterized by two anisotropic conductivity values: $\begin{matrix} {\sigma = {\begin{bmatrix} \sigma_{zz} & \sigma_{xy} & \sigma_{zy} \\ \sigma_{xz} & \sigma_{xx} & \sigma_{xy} \\ \sigma_{yz} & \sigma_{yx} & \sigma_{yy} \end{bmatrix} = \begin{bmatrix} \sigma_{V} & 0 & 0 \\ 0 & \sigma_{H} & 0 \\ 0 & 0 & \sigma_{H} \end{bmatrix}}} & \lbrack 3\rbrack \end{matrix}$

[0045] where σ_(H) is the horizontal conductivity and σ_(V) is the vertical conductivity of the formation. The formation coordinates system in this instance is (z, x, y). Where the coordinate system of the tool (l, m, n) is aligned with the coordinate system of the formation (z, x, y), the magnetic field strength in the formation can be expressed as: $\begin{matrix} {H^{formation} = \begin{bmatrix} H_{zz} & H_{zx} & H_{zy} \\ H_{xz} & H_{xx} & H_{xy} \\ H_{yz} & H_{yx} & H_{yy} \end{bmatrix}} & \lbrack 4\rbrack \end{matrix}$

[0046] However, as noted above, the borehole is rarely vertical, meaning the coordinate systems rarely align. This represents a deviation angle of θ. The borehole itself may be considered to have a coordinate system (l, t, u) where the u-axis coincides with the formation's y-axis. See FIG. 3. The borehole coordinate system and the formation coordinate system are related by a rotational operation around the y-axis by an inclination angle θ about the y-axis: $\begin{matrix} {\begin{bmatrix} \hat{l} \\ \hat{t} \\ \hat{u} \end{bmatrix} = {{\begin{bmatrix} {\cos \quad \theta} & {\sin \quad \theta} & 0 \\ {{- \sin}\quad \theta} & {\cos \quad \theta} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} \hat{z} \\ \hat{x} \\ \hat{y} \end{bmatrix}} = {{R_{y}(\theta)}\begin{bmatrix} \hat{z} \\ \hat{x} \\ \hat{y} \end{bmatrix}}}} & \lbrack 5\rbrack \end{matrix}$

[0047] where R_(y) is the rotational operator for angle θ.

[0048] Presuming that the longitudinal axis of the tool is aligned with the borehole coordinate system, then it may be stated that the antenna coordinate system (l, m, n) is aligned with the borehole coordinate system (l, t, u) and $\begin{matrix} {H^{borehole} = \begin{bmatrix} H_{ll} & H_{xy} & H_{xz} \\ H_{yx} & H_{yy} & H_{yz} \\ H_{zx} & H_{zy} & H_{zz} \end{bmatrix}} & \lbrack 6\rbrack \end{matrix}$

[0049] Moreover, H^(borehole) and H^(formation) are related by the rotational factor: $\begin{matrix} {\begin{bmatrix} H_{ll} & H_{lt} & H_{lu} \\ H_{tl} & H_{tt} & H_{tu} \\ H_{ul} & H_{ut} & H_{uu} \end{bmatrix} = {{{R_{y}(\theta)}\begin{bmatrix} H_{zz} & H_{zx} & H_{zy} \\ H_{xz} & H_{xx} & H_{xy} \\ H_{yz} & H_{yx} & H_{yy} \end{bmatrix}}{R_{y}(\theta)}^{tr}}} & \lbrack 7\rbrack \end{matrix}$

[0050] Where R_(y)(θ)^(tr) is the transposition of R_(y)(θ).

[0051] However it is rare that the coordinate system (l, m, n) of the tool antennae is aligned with the borehole coordinate system (l, t, u). Accordingly, the transverse (m, n) tool coordinates are related to the borehole coordinates (t, u) be a rotational operation about the l-axis by an azimuthal angle Φ: $\begin{matrix} {\begin{bmatrix} \hat{l} \\ \hat{m} \\ \hat{n} \end{bmatrix} = {{\begin{bmatrix} 1 & 0 & 0 \\ 0 & {\cos \quad \phi} & {\sin \quad \phi} \\ 0 & {{- \sin}\quad \phi} & {\cos \quad \phi} \end{bmatrix}\begin{bmatrix} \hat{l} \\ \hat{t} \\ \hat{u} \end{bmatrix}} = {{R_{l}(\theta)}\begin{bmatrix} \hat{l} \\ \hat{t} \\ \hat{u} \end{bmatrix}}}} & \lbrack 8\rbrack \end{matrix}$

[0052] The tool response H^(tool) is then related to the borehole response H^(borehole) be the operator $\begin{matrix} {\begin{bmatrix} H_{ll} & H_{lt} & H_{lu} \\ H_{tl} & H_{tt} & H_{tu} \\ H_{ul} & H_{ut} & H_{uu} \end{bmatrix} = {{{R_{l}(\theta)}^{tr}\begin{bmatrix} H_{ll} & H_{l\quad m} & H_{l\quad n} \\ H_{m\quad l} & H_{m\quad m} & H_{mn} \\ H_{nl} & H_{n\quad m} & H_{nn} \end{bmatrix}}{R_{l}(\phi)}}} & \lbrack 9\rbrack \end{matrix}$

[0053] where R_(l)(Φ)^(tr) is defined as the transposition of R_(l)(Φ) .

[0054] 2. Tool Response

[0055] Having defined the coordinate systems for the tool, borehole and formation, the tool response may now be expressed in terms of the formation coordinate system (z, x, y), where the tool (l, m, n) directions are aligned with the formation. At a tool transmitter, the fields in the formation can be described per Moran/Gianzero (J. H. Moran and S. C. Gianzero, “Effects of Formation Anisotropy on Resistivity Logging Measurements”, Geophysics (1979) 44, p. 1266) as follows: $\begin{matrix} {H_{zz} = {\frac{M_{z}}{4\pi}\left\{ {{\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\frac{z^{2}}{r^{2}}} - \left( {1 - u + u^{2}} \right)} \right\} \frac{e^{u}}{r^{3}}}} & \lbrack 10\rbrack \\ {H_{zx} = {\frac{M_{z}}{4\pi}\left\{ {\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\frac{xz}{r^{2}}} \right\} \frac{e^{u}}{r^{3}}}} & \lbrack 11\rbrack \\ {H_{xx} = {{\frac{M_{x}}{4\pi}\left\{ {{\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\frac{x^{2}}{r^{2}}} - \left( {1 - u + u^{2}} \right)} \right\} \frac{e^{u}}{r^{3}}} + I_{0}}} & \lbrack 12\rbrack \\ {H_{yy} = {{\frac{M_{y}}{4\pi}\left\{ {- \left( {1 - u + u^{2}} \right)} \right\} \frac{e^{u}}{r^{3}}} - I_{0} + I_{1}}} & \lbrack 13\rbrack \end{matrix}$

H _(yz)=0  [14]

H _(yx)=0  [15]

[0056] where

u=ik _(H) r  [16]

[0057] and k_(H) is the frequency of the magnetic moment induced and r is

r={square root}{square root over (x² +y ² +z ²)}; and ρ={square root}{square root over (x+ ² +y ²)}.

[0058] The receiver coils are also presumed to be located relative to the formation coordinate system (z, x, y)=(Lcosθ, Lsinθ, 0), where θ is again the deviation angle. The currents induced in the receiver coils may be expressed as: $\begin{matrix} {I_{0} = {{\frac{M_{z}}{4\pi}\left\{ {\frac{u}{r}\left( {e^{u} - e^{u\quad \beta}} \right)} \right\} \frac{1}{\rho^{2}}} = {{\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}} = {\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}{\overset{\sim}{I}}_{0}}}}} & \lbrack 17\rbrack \\ {I_{1} = {{\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}{u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} = {\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}{\overset{\sim}{I}}_{1}}}} & \lbrack 18\rbrack \end{matrix}$

[0059] where α is the inverse of λ and β is the anisotropy factor β={square root}{square root over (1+(α²−1)sin²θ)}, r=L, and ρ=Lsinθ. It should be noted that in this instance u is a function of only the horizontal resistivity. Both I₀ and I₁ are dependent on u, β and θ. If all of the transmitters are set at equal transmission power (M₁=M_(u)=M_(t)=M₀), then the tool response in the borehole can be written as: $\begin{matrix} {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left( {1 - u} \right)} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} & \lbrack 19\rbrack \\ {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left( {1 - u + u^{2}} \right)} + {u\frac{\cos^{2}\theta}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} & \lbrack 20\rbrack \\ {H_{lt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {u\frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}} & \lbrack 21\rbrack \\ {H_{uu} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left( {1 - u + u^{2}} \right)} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}} & \lbrack 22\rbrack \end{matrix}$

H _(ul)=0  [23]

H _(ut)=0  [24]

[0060] This results in three unknowns that characterize the formations anisotropic resistivity, R_(H), R_(V), and the deviation angle θ. Note that H_(ll) depends only on u (hence R_(H)) and β, only. The remainder of the responses, H_(tt), H_(lt), H_(uu) depend on the variables u, β, and θ.

[0061] 3. Azimuthal Correction for Deviated Boreholes

[0062] In deviated boreholes, the azimuthal rotation of the tool in the borehole must be determined. In actual logging, the tool's azimuthal orientation is not known. The directions of two transversally oriented antennas do not coincide with the l- and u-axis directions. The multi-component induction tool measures H^(tool) that is different from H^(borehole).

[0063] In a longitudinally anisotropic formation, the orthogonality condition holds as H_(yz)=H_(yx)=0. This implies H_(ul)=H_(ut)=0. As a result, not all six measurements of H^(tool) are independent, and (H_(lm), H_(ln), H_(mm), H_(nn), H_(mn)) must satisfy the following consistency condition: $\begin{matrix} {H_{mn} = \frac{\left( {H_{m\quad m} - H_{nn}} \right)H_{l\quad m}H_{l\quad n}}{\left( {H_{l\quad m}^{2} - H_{l\quad n}^{2}} \right)}} & \lbrack 25\rbrack \end{matrix}$

[0064] The azimuthal angle θ is determined either from (H_(lm), H_(ln)) by, $\begin{matrix} {{\tan \quad \phi} = {- \frac{H_{l\quad n}}{H_{l\quad m}}}} & \lbrack 26\rbrack \\ {{\cos \quad \phi} = \frac{H_{l\quad m}}{\sqrt{H_{l\quad m}^{2} + H_{l\quad n}^{2}}}} & \lbrack 27\rbrack \\ {{\sin \quad \phi} = {- \frac{H_{l\quad n}}{\sqrt{H_{l\quad m}^{2} + H_{l\quad n}^{2}}}}} & \lbrack 28\rbrack \end{matrix}$

[0065] or from (H_(mm), H_(nn), H_(mn)) by, $\begin{matrix} {{\tan \quad 2\phi} = {{- \frac{2H_{mn}}{H_{m\quad m} - H_{nn}}} = \frac{2\tan \quad \phi}{1 - {\tan^{2}\phi}}}} & \lbrack 29\rbrack \end{matrix}$

[0066] If all of the (H_(lm), H_(ln), H_(mm), H_(nn), H_(mn)) measurements are available, the azimuthal angle θ may be determined by minimizing the error, $\begin{matrix} {{error} = {{{{H_{l\quad m}\sin \quad \phi} + {H_{l\quad n}\cos \quad \phi}}}^{2} + {{{\frac{H_{m\quad m} - H_{nn}}{2}\sin \quad 2\phi} + {H_{mn}\cos \quad 2\phi}}}^{2}}} & \lbrack 30\rbrack \end{matrix}$

[0067] The H^(borehole) is calculated in terms of H^(tool) by, $\begin{matrix} {H_{lt} = \sqrt{H_{l\quad m}^{2} + H_{l\quad n}^{2}}} & \lbrack 31\rbrack \\ {H_{tt} = \frac{{H_{m\quad m}H_{l\quad m}^{2}} - {H_{nn}H_{l\quad n}^{2}}}{H_{l\quad m}^{2} - H_{l\quad n}^{2}}} & \lbrack 32\rbrack \\ {H_{uu} = \frac{{H_{nn}H_{l\quad m}^{2}} - {H_{m\quad m}H_{l\quad n}^{2}}}{H_{l\quad m}^{2} - H_{l\quad n}^{2}}} & \lbrack 33\rbrack \end{matrix}$

[0068] These should be used in determining the R_(H), R_(V), and θ, from the multi-component induction tool measurements H^(tool).

[0069] 4. Inversion of Multi-Component Induction Resistivity Data

[0070] The present invention utilizes an inversion technique to determine anisotropic resistivity characteristics over a range of deviation angles. Where the tool is placed in vertical borehole (θ=0), there exist two independent measurements, each independent measurement being composed of an H field having both an in-phase and out-phase component. $\begin{matrix} {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {2\left( {1 - u} \right)} \right\}}} & \lbrack 34\rbrack \\ {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {- \left( {1 - u + {u^{2}\alpha^{2}} + \frac{1}{2}} \right)} \right\}}} & \lbrack 35\rbrack \end{matrix}$

H _(lt)=0  [36]

H _(uu) =H _(tt)  [37]

[0071] This means that H_(ll) is a function solely of u, hence the horizontal resistivity. H_(tt)=H_(uu) which means both are a function of u and anisotropy λ²=R_(H)/R_(V), i.e., a function of both the horizontal and vertical resistivity.

[0072]FIG. 3A demonstrates the relationship between the relationship between the R-signal and the formation horizontal resistivity R_(H). This relationship may be used to invert the conventional R signal to obtain apparent R_(H). When using both the resistive R and reactive X components of the signal measured as part of H_(ll), the graph set forth in FIG. 3B may be used to determine R_(H). This is done by minimizing the model error as follows: $\begin{matrix} {{error} = {{H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {2\left\lbrack {1 - u} \right\rbrack} \right\}}}}^{2}} & \lbrack 38\rbrack \end{matrix}$

[0073] Once R_(H) is determined from the above, the vertical resistivity R_(V) can be determined from the H_(ll) measurement. One means of doing so is to determine the ratio R_(H)/R_(V) as a function of the R-signal as set forth in FIG. 3C. An alternative means of doing so would be based on the ratio of the measured H_(tt)/H_(ll), ratio based again on the resistive component of the signal R, i.e., R_(tt)/R_(ll), as demonstrated in FIG. 3D.

[0074] An alternative means of determining horizontal and vertical resistivity may be accomplished utilizing both the resistive and reactive portions of the received signal from H_(tt). FIG. 4A is a nomograph showing differing ratios of anisotropic resistivity values R_(H), R_(V) (in this instance, as a function of λ, the square root of R_(H)/R_(V)) . Both R_(H) and R_(V) can be determined simultaneously by minimizing the error as follows: $\begin{matrix} {{error} = {{H_{tt}^{measured} + {\frac{M_{0}}{4\pi}{\frac{e^{u}}{r^{3}}\left\lbrack {1 - u + {u^{2}\alpha^{2}} + \frac{1}{2}} \right\rbrack}}}}^{2}} & \lbrack 39\rbrack \end{matrix}$

[0075] An alternative means to determine R_(H) and R_(V) is shown in nomograph of FIG. 4B, which shows R_(V)/R_(H) as a function of both the R and X signals from H_(tt). Both R_(H) and R_(V) can be determined simultaneously by again minimizing Eq. 29.

[0076] When the tool records R- and X-signals for both H_(ll) and H_(tt), R_(H) and R_(V) can be determined simultaneously by minimizing the error: $\begin{matrix} {{error} = \left| {H_{ll}^{measured} - {\frac{M_{0}e^{u}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {2\left\lbrack {1 - u} \right\rbrack} \right\}}} \middle| {}_{2}\quad {+ {\quad\left| {H_{tt}^{measured} + {\frac{M_{0}}{4\pi}{\frac{e^{u}}{r^{3}}\left\lbrack {1 - u + {u^{2}\alpha^{2}} + \frac{1}{1}} \right\rbrack}}} \right|^{2}}} \right.} & \lbrack 40\rbrack \end{matrix}$

[0077] As noted previously, it is rare in current drilling and logging practice that a well is drilled vertical. At the opposite end of the spectrum is a determination of anisotropic resistivity characteristics when the borehole is essentially horizontal (θ=90°). Equations 19-24 reduce to three independent measurements, each with a resistive and reactive component: $\begin{matrix} {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left( {1 - u} \right)} + {u\left( {1 - e^{u{({\alpha - 1})}}} \right)}} \right\}}} & \lbrack 41\rbrack \\ {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {- \left( {1 - u + u^{2}} \right)} \right\}}} & \lbrack 42\rbrack \\ {H_{uu} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left( {1 - u + u^{2}} \right)} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} + {u^{2}\left( {1 - {\alpha \quad e^{u{({\alpha - 1})}}}} \right)}} \right\}}} & \lbrack 43\rbrack \end{matrix}$

H _(lt) =H _(ul) =H _(ut)=0  [44]

[0078] Herein, H_(tt) is a function solely of u, and hence is a function solely of horizontal resistivity R_(H). H_(tt) and H_(uu) are both a function of variables u and α, i.e., u and R_(H) and R_(V).

[0079]FIG. 5A is a nomograph that relates the R signal from H_(tt) to the horizontal resistivity R_(H). This may be used to invert the R signal to the apparent formation resistivity. Where R and X signals are available for H_(tt), they may also be used to obtain a formation horizontal resistivity for differing R_(H) as shown in nomograph 5B. This is accomplished my minimizing the error function: $\begin{matrix} {{error} = \left| {H_{tt}^{measured} + {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {1 - u + u^{2}} \right\}}} \right|^{2}} & \lbrack 45\rbrack \end{matrix}$

[0080] Once the R_(H) is determined by either method above, the anisotropy and the vertical resistivity can be determined from various signals based on the H_(ll) measurement. The anisotropy R_(V)/R_(H) may be obtained from the R signal based on the H_(ll) measurement as shown in FIG. 5C. Alternatively, the anisotropy R_(V)/R_(H) may be obtained as a function of the R signal from H_(uu) as shown in FIG. 5D. It should be noted that a determination using FIG. 5D is more highly dependent on R_(H) as opposed to R_(V) at high ohm-m resistivities. A more accurate determination of R_(V)/R_(H) may be made as a ratio of the R signals from H_(uu)/H_(ll) as demonstrated in FIG. 5E. Another inversion that is useful only at low resistivities is depicted in FIG. 5E which attempts to derive R_(V)/R_(H) as a function of the ratio R signals received an antennae H_(uu)/H_(tt).

[0081] In the preferred method of the present invention, both the R and X signals are used to determine R_(H) and R_(V) simultaneously. In the nomograph of FIG. 6A, R_(V) and R_(H) are determined as a function of the ratio of the R/X signals at H_(ll) is obtained for differing anisotropic values. R_(H) and R_(V) may be determined by minimizing the error function: $\begin{matrix} {{error} = \left| {H_{tt}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\alpha - 1})}}} \right)}} \right\}}} \right|^{2}} & \lbrack 46\rbrack \end{matrix}$

[0082] However, the ratio of R_(V)/R_(H) may be better determined from the ratio X/R signals received at H_(ll) as demonstrated in FIG. 6B. As in FIG. 6A, one may determined R_(V) and R_(H) for differing anisotropy values based on the ratio of the R/X signals received at H_(uu), as shown in FIG. 6C and then minimizing the following error function to determine R_(V) and R_(H): $\begin{matrix} {{error} = \left| {H_{uu}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} + {u^{2}\left( {1 - {\alpha e}^{u{({\alpha - 1})}}} \right)}} \right\}}} \right|^{2}} & \lbrack 47\rbrack \end{matrix}$

[0083] As with the signals received at H_(ll), a better determination of the ratio R_(V)/R_(H) may be made with respect to anisotropy values based on the ratio of X/R signals received at H_(uu) as demonstrated in FIG. 6D and then applying the error function.

[0084] When both R and X signals are available from all antennae locations H_(ll), H_(uu), and H_(tt), R_(H) and R_(V) may be determined simultaneously with greater accuracy by minimizing the error function: $\begin{matrix} \begin{matrix} {{error} = \quad \left. \left| H_{tt}^{measured} + {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {1 - u + u^{2}} \right\}} \right. \middle| {}_{2} + \right.} \\ {\quad \left| {H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \middle| {}_{2} + \right.} \\ {\quad \left| {H_{uu}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} + {u^{2}\left( {1 - {\alpha \quad e^{u{({\beta - 1})}}}} \right)}} \right\}}} \right|^{2}} \end{matrix} & \lbrack 48\rbrack \end{matrix}$

[0085] The two above methods address determination of R_(H) and R_(V) at the most extreme cases, i.e. θ=0° or 90°. More often than not, the deviation angle for the borehole will be somewhere within this range. The above method for determining the formation vertical and horizontal resistivities may be used where the deviation angle is less than 30°. However the effect of the deviation angle becomes significant for values above 30°. In which instance, the full form of Eqs. 19-24 must be used: $\begin{matrix} {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \\ {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} + {u\frac{\cos^{2}\theta}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \\ {H_{lt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {u\frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}} \\ {H_{uu} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}} \\ {H_{ul} = {H_{ut} = 0}} \end{matrix}$

[0086] There are three variables that characterize the formation's anisotropic characteristics. H_(ll) depends only on u (hence, R_(H)) and β, whereas, H_(tt), H_(lt), and H_(uu) are all dependent on u, β, and θ. Herein there are four independent measurements, each having an R and X component, for this overly constrained model.

[0087] Using both the R and X signals from H_(ll) can be used to determine u (or R_(H)) and β, if both θ (recalling that β={square root}{square root over (cos²θ+α²sin²θ)} and as θ→90°, β→α) and R_(V)/R_(H) are large. This relationship is demonstrated in nomograph FIG. 7A, from which one may determining R_(H) and β as a function of the R and X signals from H_(ll) for varying R_(H) and β. R_(H) and β may also be determined from the R and X signals of H_(ll) by minimization of the error function: $\begin{matrix} {{error} = \left| {H_{ll} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \right|^{2}} & \lbrack 49\rbrack \end{matrix}$

[0088] Alternatively, R_(H) may be determined from the ratio of the R/X signals at H_(ll) as shown in nomograph FIG. 7A. Upon determining R_(H) (or u) and β, the may be substituted into Eqs. 20-22 to determine the remaining two variables, R_(V) and θ. Thus, one can determine the horizontal and vertical resistivities, R_(H) and R_(V), without prior knowledge of the deviation angle θ, which may be independently determined.

[0089] When R and X signals are available from H_(ll), H_(tt), H_(lt), and H_(uu), then R_(H) and R_(V) and θ may be determined by minimizing the error function: $\begin{matrix} \begin{matrix} {{error} = \quad \left| {H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \middle| {}_{2} + \right.} \\ {\quad \left. {H_{tt}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} + {u\frac{\cos^{2}\theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \middle| {}_{2} + \right.} \\ {\quad \left. {H_{lt}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {u\frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}} \middle| {}_{2} + \right.} \\ {\quad \left. {H_{uu}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}} \right|^{2}} \end{matrix} & \lbrack 50\rbrack \end{matrix}$

[0090] Commercially available computer programs may be used to minimize the error function of Eq. 50. The minimization of the error as defined in Eq. 50 permits the use of measured values, as opposed to values that have been rotated through the azimuth and dip corrections, to simultaneously determine R_(H), R_(V) and θ.

[0091] Note that if the deviation angle θ is already known, and is not small, then the any or all of the Eqs. 20-22 may be used to determine R_(H) and R_(V). For instance, both the R and X signals from H_(ll) can be used to determine R_(H) and β, then R_(V) from β if angle θ is known.

[0092] Accordingly, the preferred embodiment of the present invention discloses a means for determining earth formation anisotropic resistivity utilizing a multi-component induction tool. Moreover, a method is disclosed for performing inversion techniques to determine the said characteristics utilizing various combinations of R and X signals at varying antenna locations. Moreover, a method of determining formation deviation or dip angle has been disclosed.

[0093] While the invention is susceptible to various modifications and alternative forms, specific embodiments have been shown by way of example in the drawings and that have been described in detail herein. However, it should be understood that the invention is not intended to be limited to the particular forms disclosed. Rather, the invention is to cover all modifications, equivalents, an alternatives falling within the spirit and scope of the present invention as defined by the appended claims. 

What is claimed:
 1. A method for determining the horizontal resistivity and vertical resistivity of an earth formation, the earth formation being penetrated by a borehole, comprising: (a) developing an inversion model for various earth formations; (b) deploying an induction tool in said borehole, said tool having a longitudinal axis, a transmitter array comprised three mutually orthogonal transmitter antennae, at least one of the antenna being oriented parallel to said tool longitudinal axis, and a receiver array offset from the transmitter array, said receiver array being comprised of three mutually orthogonal receiver antennae, the receiving array sharing a common orientation with said transmitter array; (c) activating said transmitter antenna array and measuring electromagnetic signals values induced in said receiving array antennae, including resistive and reactive components of said signal values; (d) determining an azimuth angle for said tool; (e) calculating secondary signal values signals as a function of said measured signal values and said azimuth angle; (f) determining a dip angle for said tool; (g) calculating tertiary signal values as a function of said secondary signal values and said dip angle; (h) calculating said horizontal resistivity as a function of said resistive and reactive components of said tertiary signal values; and (i) calculating an anisotropy parameter as a function of said resistive and reactive components of said tertiary signal values and determining said vertical resistivity as a function of said anisotropy parameter and said horizontal resistivity.
 2. The method of claim 1, wherein said azimuth angle is calculated as a function of said electromagnetic signals in said receiver array antennae perpendicular to said tool longitudinal axis, relative to a direction of said horizontal resistivity and said vertical resistivity.
 3. The method of claim 1, wherein said dip angle is calculated as a function of said secondary values for said electromagnetic signals for a receiver antenna parallel to said tool longitudinal axis with respect to said vertical resistivity.
 4. The method of claim 1, wherein said secondary signal values are calculated by rotating said measured signals values through a negative of said azimuth angle.
 5. The method of claim 1, wherein said tertiary signal values are calculated by rotating said secondary signal values through a negative of said dip angle.
 6. The method of claim 1, wherein said horizontal resistivity is calculated as a function of said resistive and reactive components of said tertiary signal values measured in a selected receiving array antenna, where said antenna is parallel to said tool longitudinal axis.
 7. The method of claim 1, wherein said horizontal resistivity is calculated as a function of said resistive and reactive components of said tertiary signal values measured in a selected receiving array antenna, where said antenna is in a plane perpendicular to said tool longitudinal axis.
 8. The method of claim 1, wherein said anisotropy parameter is calculated as a function of said horizontal resistivity and said resistive components of said tertiary signal values measured in at least two selected receiving array antennae, where one antenna is parallel to said to said tool longitudinal axis and at least one antenna is in a plane perpendicular to said tool longitudinal axis.
 9. The method of claim 1, wherein said anisotropy parameter is calculated as a function of said horizontal resistivity and said resistive and reactive components of said tertiary signal values measured in a selected receiving array antenna, where said selected antenna is in a plane perpendicular to said tool longitudinal axis.
 10. A method for determining the horizontal resistivity and vertical resistivity of an earth formation, the earth formation being penetrated by a borehole, comprising: (a) developing an inversion model for various earth formations; (b) deploying an induction tool in said borehole, said tool having a longitudinal axis, a transmitter array comprised three mutually orthogonal transmitter antennae, at least one of the antenna being oriented parallel to said tool longitudinal axis, and a receiver array offset from the transmitter array, said receiver array being comprised of three mutually orthogonal receiver antennae, the receiving array sharing a common orientation with said transmitter array; (c) activating said transmitter antenna array and measuring electromagnetic signal values induced in said receiving array antennae, including resistive and reactive components of said signal values; and (d) simultaneously determining said horizontal resistivity, vertical resistivity and a dip angle by as a function of selected resistive and reactive components of said signal values measured in said receiving array antennae.
 11. The method of claim 10, wherein said selected resistive and reactive components of said measured signal values in selected receiving array antennae are induced by selected transmitter array antennae.
 12. The method of claim 11, wherein said selected resistive and reactive components of said measured signal values in said receiving antennae array are induced by transmitter array antennae similarly oriented and a non-zero signal induced by said transmitter array antenna oriented parallel to said tool longitudinal axis, measured in a receiver array antenna lying in a plane perpendicular to said transmitter array antenna oriented parallel to said tool longitudinal axis. 